Model of damage of MAZARS - Code_Aster · 2019. 11. 23. · Code_Aster Version default Titre : Modèle d’endommagement de MAZARS Date : 25/09/2013 Page : 3/16 Responsable : HAMON

Code_Aster VersiondefaultTitre : Modèle d’endommagement de MAZARS Date :

25/09/2013Page : 1/16

Responsable : HAMON François Clé : R7.01.08 Révision :208cf76e4941

Model of damage of MAZARS

Summary:

This documentation presents the model of behavior of MAZARS who allows to describe the behaviorrubber band-endommageable of the concrete. This model is 3D, isotropic and is based on a criterion ofdamage written in deformation and describing dissymmetry traction and compression. The initial model,does not give an account of the restoration of rigidity in the event of “refermeture of the cracks” anddoes not take into account the possible plastic deformations or viscous effects which can be observedduring deformations of a concrete. The version implemented in Code_Aster takes account of the lastimprovements. This reformulation of the Mazars model of the years 1980 makes it possible to betterdescribe the behavior of the concrete in bi--compression and pure shearing.The version 1D model makes it possible to give an account of the restoration of rigidity in the event ofrefermeture of the cracks.

Three versions of the model are established:• the local version (with risk of dependence to the grid)• the not-local version where the damage is controlled by the gradient of deformation. It is also

possible to take into account the dependence of the parameters of the law with the temperature, thehydration and drying.

• the version 1D local, only used with the multifibre beams [R5.03.09].

Warning : The translation process used on this website is a "Machine Translation". It may be imprecise and inaccurate in whole or inpart and is provided as a convenience.Copyright 2019 EDF R&D - Licensed under the terms of the GNU FDL (http://www.gnu.org/copyleft/fdl.html)


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Contents1 Introduction................................................................................................................................ 3

1.1 A law of behavior élasto-endommageable..........................................................................31.2 Limits of the local approach and methods of regularization................................................31.3 Coupling with thermics........................................................................................................41.4 Law of Mazars in the presence of a field of drying or hydration..........................................4

2 Models of MAZARS...................................................................................................................62.1 Model of Origin of Mazars...................................................................................................62.2 Revisited model of Mazars..................................................................................................7

3 Identification............................................................................................................................ 104 Digital resolution...................................................................................................................... 13

4.1 Evaluation of the internal variable Y.................................................................................134.2 Evaluation of the damage.................................................................................................134.3 Calculation of the constraint..............................................................................................134.4 Calculation of the tangent matrix......................................................................................134.5 Stored internal variables...................................................................................................14

5 Features and checking.............................................................................................................146 Bibliography............................................................................................................................. 157 History of the versions of the document..................................................................................16



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1 Introduction1.1 A law of behavior élasto-endommageable

The model of behavior MAZARS 1] is a model simple, considered robust, based on themechanics of the damage [feeding-bottle2] , which makes it possible to describe the reductionin the rigidity of material under the effect of the creation of microscopic cracks in the concrete.It is based on only one variable interns scalar D describing the isotropic damage of way, butdistinguishing despite everything the damage from traction and the damage from compression.The version implemented under Aster corresponds to the reformulation of 2012 [feeding-bottle3] . Major modification compared to the model of origin 1] is the introduction of a newinternal variable, noted Y , corresponding to the maximum reached during the loading by theequivalent deformation defined in the years 1980. So the damage is not any more the internalvariable in the revisited model. Moreover, its law of evolution is simplified in order to eliminatethe concepts of damage from traction and compression.

Contrary to the model ENDO_ISOT_BETON, this model does not make it possible to translatethe phenomenon of refermeture of the cracks (restoration of rigidity). In addition, the model ofMAZARS does not take into account the possible plastic deformations or viscous effects whichcan be observed during deformations of a concrete.

The version 1D model of MAZARS is described in the document [R5.03.09] “non-linearRelations of behavior”. In this specific case, the model is able to give an account of thephenomenon of refermeture of the cracks. The version 1D model, is usable only with themultifibre beams.

1.2 Limits of the local approach and methods of regularization Like all the lenitive laws, the model of MAZARS raise difficulties related to the phenomenon oflocalization of the deformations.

Physically, the heterogeneity of the microstructure of the induced concrete of the remoteinteractions enters the formed cracks 4]. Thus, the deformations locate in a metal strip, calledband of localization, to form the macro-cracks. The state of the constraints in a material pointcannot be any more only described by the characteristics at the point but must also take intoaccount its environment. In the case of the local model, no indication is included concerning thescale of cracking. Consequently, no information is not given over the bandwidth of localizationwhich becomes worthless then. This leads to a mechanical behavior with rupture withoutdissipation of energy, physically unacceptable.

Mathematically, the localization returns the problem to be solved badly posed becausesoftening causes a loss of ellipticity of the differential equations which describe the process ofdeformations 5]. The digital solutions do not converge towards physically acceptable solutionsin spite of refinements of grid.Numerically, one observes a dependence of the solution to the network extremely prejudicial (cf[R5.04.02]).

A method of regularization thus becomes necessary. Several are possible. The choice whichwas made here is to regularize in gradient of deformation, and to thus use a tensor ofregularized deformation who checks the characteristic equation [R5.04.02]:

=−Lc2 ∇2 (éq 1.2-1)

where the scalar Lc (characteristic length) the dimension a length is.

Note:



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Let us announce that this model not-room does not correspond to the version initiallyproposed by J.Mazars and G.Pijaudier-Pooch 6] and which is in particular established inCAST3M, where delocalization is obtained while using as equivalent deformation, the averageof the deformation equivalent local on a volume V :

x = 1V r x ∫

x−s eq s ds

where is the volume of the structure

V rx is representative volume at the point x : V r=∫

x−s ds

x−s is a weight function: x−s =exp−4∥x−s∥2l c2 l c is an internal length (traditionally estimated at three times size of the largest

aggregate).

Digital tests made it possible to connect the 2 parameters of delocalization l c and Lc in thecase of the model of Mazars. The following relation was obtained: 4Lc≃l c

The model of MAZARS is thus available in Code_Aster under 2 versions:• the local version of the model for which the dependence of the solution to the network is

observable as for all the lenitive models.• a nonlocal version which uses a tensor of regularized deformation (known as also

“nonlocal”), modeling of the type GRAD_EPSI.

1.3 Coupling with thermicsFor certain studies, it can be interesting to be able to take into account the modification of theparameters materials under the effect of the temperature. This is possible in Aster (MAZARS_FOcompound or not with ELAS_FO). The assumptions made for the coupling with thermics are thefollowing ones:• thermal dilation is supposed to be linear is:

th= T−T ref Id (éq 1.3-1)with α = constant or function of the temperature,

• one does not take into account thermomechanical interactions, i.e. one does not model theeffect of the mechanical state of stress on the thermal deformation of the concrete,

• concerning the evolution of the parameters materials with the temperature, one considersthat those depend not on the current temperature but on the maximum temperature Tmaxsight by material during its history, (effect quoted in the literature),

• only elastic strain (mechanical) induced of the damage.

Note:Because of data-processing constraints, the initial value of Tmax is initialized to 0.Consequently, one cannot use the parameters materials defined for negative temperatures (ifnecessary, one can however circumvent this problem while returning all the temperatures inKelvin instead of °C ).

1.4 Law of Mazars in the presence of a field of drying or hydrationThe use of ELAS_FO and/or MAZARS_FO under the operator DEFI_MATERIAU allows to makedepend the parameters materials on drying or hydration.



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In addition, deformations related to the withdrawal of endogenous re and with the withdrawaldesiccation rd are taken into account in the model, in the form (linear) following (cf.[R7.01.12]) :

re=− Id (éq 1.4-1)

rd=−C ref−C Id (éq 1.4-2)where is the hydration, C water concentration (field of drying in the terminologyCode_Aster), C ref initial water concentration (or drying of reference). Finally is theendogenous coefficient of withdrawal and the coefficient of withdrawal of desiccation to beinformed in DEFI_MATERIAU, keyword factor ELAS_FO, operands B_ENDO and K_DESSIC. Asone said to the preceding paragraph, the choice which was made in the establishment of themodel of MAZARS, it is that only the elastic strain induced of the damage. Consequently, if onemodels a concrete test-tube which dries or which is hydrated freely and uniformly, one willobtain well a field of deformation not no one and a stress field perfectly no one.

One initially presents the writing of the model then some data on the identification of theparameters. To finish, one exposes the principles of digital integration in Code_Aster.



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2 Models of MAZARS2.1 Model of Origin of Mazars

The model of MAZARS was elaborate within the framework of the mechanics of the damage.This model is detailed in the thesis of MAZARS 1]The constraint is given by the following relation:

=1−D E e (éq 2.1-1)with : • E the matrix of Hooke, • D the variable of damage• e elastic strain e=− th− rd− re

• th= T−T ref Id thermal dilation• re=−Id endogenous withdrawal (related to the hydration)• rd=−C ref−C Id withdrawal of desiccation (related to drying)

D is the variable of damage. It is understood enters 0 , materials healthy, and 1 , brokenmaterial. The damage is controlled by the equivalent deformation eq who allows to translate atriaxial state by an equivalence in a uniaxial state. As the extensions are paramount in thephenomenon of cracking of the concrete, the introduced equivalent deformation is definedstarting from the positive eigenvalues of the tensor of the deformations, that is to say:

eq=〈 〉: 〈 〉 where in the principal reference mark of the tensor of deformations:

eq= 〈1〉2〈2〉2〈3〉2 (éq 2.1-2)

knowing that the positive part 〈 〉 is defined so that if i is the principal deformation in thedirection i :

{〈i〉=i si i≥0〈i〉=0 si i0 (éq 2.1-3)Note:In the case of a thermomechanical loading, only elastic strain e=− th contribute tothe evolution of the damage from where: eq=〈 e〉 : 〈 e〉 .

eq is an indicator of the state of tension in the material which generates the damage. This sizedefines the surface of load f such as:

f =eq−K D =0 (éq 2.1-4) (4)

with K D =d0 if D=0 . d0 the deformation threshold of damage. When the equivalent deformation reaches this value, the damage is activated. D is defined likea combination of two damaging modes defined by Dt and Dc , variable between 0 and 1depending on the state of associated damage, and corresponding respectively to the damage intraction and compression. The relation binding these variables is the following one:

D=t Dtc

D c (éq 2.1-5) (5)



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is a coefficient which was introduced later on to improve behaviour in shearing. Usually itsvalue is fixed at 1.06. Coefficients t and c carry out a link between the damage and thecompactness of traction or. When traction is activated t =1 whereas t =0 andconversely in compression. A characteristic of this model is its explicit writing what implies that all the sizes are calculateddirectly without using an algorithm of linearization like that of Newton-Raphson. Thus, laws ofevolution of the damages Dt and Dc express themselves only starting from the equivalentdeformation eq

Dt=1−1−At d0

eq−Atexp −Bt eq−d0 (éq 2.1-6)

D c=1−1−Ac d0

eq−Ac exp −Bc eq−d0 (éq 2.1-7)

with At , Ac , Bt , and Bc , parameters materials to be identified. These parameters make itpossible to modulate the shape of the curved post-peak. They are obtained using test andtensile tests of compression.

2.2 Revisited model of MazarsAlthough usually employed, the model of Origin of Mazars has gaps in the modeling of thebehavior of the concrete during loadings in shearing and bi--compression. A comparisonbetween surfaces of load of the two models is given in Figure 2.2-4.

a) Model of Origin of Mazars b) Revisited model of Mazars

Figure 2.2-1 : Comparison of surfaces of initiation of damage and rupture of the Mazarsmodels in the plan 3=0 and a C30 concrete

Thus, a new formulation is proposed through 2 major modifications:

1. improvement of behaviour in bi--compression,2. simplification and improvement of behaviour in shearing.

The model of Mazars of origin of the years 1980 1] the resistance of the concrete in bi--compression underestimates largely. The first modification made by the Revisited model thusimproves behaviour in bi--compression. This goal is reached by correcting the equivalentdeformation when at least a principal constraint is negative, using one variable :



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eqcorrigée= eq= 〈 〉 :〈 〉 (éq 2.2-1)

with:

{=−∑i 〈 i〉−2

∑i〈 i 〉−

si au moins une contrainte effective est négative

=1 sinon

(limited between 0 and 1) (éq 2.2-2)

Lforced effective hasU direction of the mechanics of the damage is defined by:

=

1−D (éq 2.2-3)

The definition of < > - is similar to (éq 2.1-3) :

{〈 i〉−= i si i≤0〈 i〉−=0 si i0 (éq 2.2-4)where i is a principal effective constraint.

The improvement of behaviour in shearing is reached by the introduction of a new internalvariable: Y . It corresponds to the maximum reached during the loading of the equivalentdeformation. Its initial value Y 0 is d0 . Y is defined by the following equation:

Y=max d0 ,max eqcorrigée (éq 2.2-5)

The function of load is:

f =eqcorrigée−Y (éq 2.2-6)

The evolution of the damage is given by:

D=1−1−A Y 0

Y−A exp −B Y−Y 0 (éq 2.2-7)

In this expression, they are the variables A and B who allow to reproduce the quasi fragilebehavior of the concrete in traction and the behavior hammer-hardened in compression. Torepresent as well as possible the experimental results, the following laws of evolution wereselected for A and B :

A=At 2r 2 1−2k −r 1−4k Ac 2r2−3r1 (éq 2.2-8)

and

B=r 2B t1−r 2 Bc (éq 2.2-9)

where the expression of r is:

r=∑i〈 i 〉

∑i∣ i∣

(éq 2.2-10)



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It appears in these equations a new variable r who informs us about the state of stress.When r is equal to 1 (corresponding to the sector of tractions), the variables A and B areequivalent to the param beings At and Bt . Therefore, (éq 2.2-7) is identical to (éq 2.1-6) .Conversely, if r is worthless (corresponding to the sector of compressions), then A=Ac ,B=Bc and (éq 2.2-7) is identical to (éq 2.1-7) .

Figure 2.2-2 give in the plan 3=0 evolution according to the sign of the principalconstraints variables A , B , r and .

Figure 2.2-2 : Evolution of the variables A , B , r and in the plan 3=0

In the equation (éq 2.1-6) a new parameter appears: k . It introduces an asymptote with thecurve − in shearing and it is defined by:

k=Acisaillement

At (éq 2.2-11)

where Acisaillement the residual stress in pure shearing defines. It is similar to At for thiscase of loading. The value advised for k is of 0.7 . The value of k lower than 1 is veryuseful the modeling of the effects of friction enters the concrete and the reinforcements inreinforced concrete structures because it induces a residual shear stress. For the valuek=1 one finds the behavior of the model of Origin (Figure 2.2-3).

Figure 2.2-3 : Stress-strain curve during a pure shear test on a point of Gauss

The model of Origin under-Estime the resistance of the concrete in pure shearing. Thisnew formulation makes it possible to increase this resistance in pure shearing passing


Asymptote induced by K 1


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from 2.5MPa with 3.5Mpa for a C30 concrete. This value depends on those of theparameters materials entered ( At , Ac , Bt , and Bc ).

The local answer of the Revisited model of Mazars under loading successive of tractioncompression is given by Figure 2.2-4.

Figure 2.2-4 : Answer stress-strain of the model of Mazars for one request 1D.

Figure 2.2-4 allows to visualize a certain number of characteristics of the model ofMAZARS, namely:• the damage affects the rigidity of the concrete,• there are no unrecoverable deformations,• the answers in traction and compression are quite dissymmetrical,

Notice : The models Mazars d' Origine and Revisited do not take into account the unilateralcharacter of the concrete to knowing refermeture of crack at the time of the passage of astate of traction in a state of compression.

3 IdentificationIn addition to the thermoelastic parameters E , , , the model of Revisited MAZARS factof intervening 6 parameters material: Ac , B c , At , B t , d0 , k .• d0 is the threshold of damage. It acts obviously on the constraint with the peak but also

on the shape of the curved post-peak. Indeed, the fall of constraint is of as much lessbrutal than d0 is small. In general d0 is understood in 0.5 and 1.510−4 .

Coefficients A and B allow to modulate the shape of the curved post-peak. They are definedby the equations (éq 2.2-8) and (éq 2.2-9) who depend on the parameters of the model ofOrigin of Mazars ( At , Bt , Ac and Bc ) and of r :• A introduced a horizontal asymptote which is the axis of for A=1 and the horizontal

one passing by the peak for A=0 (cf [Figure 3-1]). In the field as of tractions, A isWarning : The translation process used on this website is a "Machine Translation". It may be imprecise and inaccurate in whole or inpart and is provided as a convenience.Copyright 2019 EDF R&D - Licensed under the terms of the GNU FDL (http://www.gnu.org/copyleft/fdl.html)


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equivalent to At (and reciprocally in the field of compressions A=Ac ). In general, Aclies between 1 and 2. and At between 0.7 and 1.

• B according to its value can correspond to a sharp fall of constraint ( B10000 ) or apreliminary phase of increase in constraint followed, after passage by a maximum, of amore or less fast decrease as one can see it on [3-2]. In the field as of tractions, B isequivalent to Bt (and reciprocally in the field of compressions B=Bc ). In general Bcis understood enters 1000 and 2000 and Bt enter 9000 and 21000 .

• k introduced a horizontal asymptote in pure shearing on stress-strain curve if its valueis different from 1 for At=1 , (éq 2.2-11). The advised value is 0.7 .

Figure 3-1: Influence of the parameter At



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FigRUE 3-2: Influence of the parameter Bt

A means of obtaining a set of parameters is to have the uniaxial test results in compressionand traction (for traction one can use other type of tests, of the “Brazilian” tests of splitting forexample). If one uses the regularization in gradient of deformation (see §1.2), it is recommended to fixthe parameters of the law at the same time characteristic length Lc . Some authors (confer[feeding-bottle7]) also suggest gauging Lc in using experimental tests on several sizes of thespecimens; indeed, the characteristic length is dependent in keeping with the energy zone ofdissipation which could be at the origin of scale effect structural.



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4 Digital resolution

4.1 Evaluation of the internal variable YThe calculation of Y is very simple and follows an explicit diagram. The stages are thefollowing ones:• Calculation of the elastic strain and thermal • Calculation of the principal elastic constraints and evaluation of (éq 2.2-2).• Calculation of the equivalent deformation ((éq 2.1-2) and (éq 2.2-1)).• Calculation of the variables r , A and B• Calculation of the internal variable Y (éq 2.2-5).

If Y≤Y− then Y=Y− .If YY− then Y=Y .

Note: Currently the variable stored during calculations is eq in order not to modify them existing couplings with UMLV. A condition on the strictly increasing evolution of the damage

allows this simplification if vary.

4.2 Evaluation of the damageThe damage is calculated in all the cases with the equation (éq 2.2-7).

D=1−1−A Y 0

Y−A exp −B Y−Y 0 (éq 2.2-7)

It is important to specify that we impose on D to be ranging between 0 and 1 because it ispossible to have values outwards this framing following the choice of the parameters materialsas for the model of origin.

4.3 Calculation of the constraintAfter evaluation of D , we calculate simply:

=1−D A e (éq 4.3-1)

4.4 Calculation of the tangent matrixOne of the disadvantage of the model of Mazars is the absence of tangent matrix. It is notpossible to calculate this matrix because of the use of the MacCauley operator in thecalculation of the equivalent deformation (éq 2.1-2), of (éq 2.2-2) and r . However it ispossible to use a radial approximation during loading.Us let us seek the tensor M such as ̇=M ̇ knowing that =1−D A . The matrix isthus the sum of two terms, one with constant damage, the other due to the evolution of thedamage:

̇=1−D A ̇−A Ḋ (éq 4.4-1)The first term is easy, it acts of the operator of Hooke, multiplied by the factor 1−D .The second requires the evaluation of the increment of damage Ḋ .If a radial loading is imposed, variables , r , A and B are constant. While posing:

Ḋ=∂D∂Y

∂Y∂ eq

∂ eq ∂

̇ (éq 4.4-2)

With



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∂Y∂ eq

∂ eq ∂

=〈 〉ε eq

(éq 4.4-3)

Under this condition of radial loading, the increment of deformation is written:

Ḋ=[ 1−A Y 0Y 2 AB exp B Y−Y 0 ] 〈〉 eq ̇ (éq 4.4-4) Note: 1. Being given made simplifications, in the case general the tangent matrix is not

consistent. Also, it can happen that the reactualization of the tangent matrix duringiterations of Newton does not help with convergence. In this case, it is enough to useonly the secant matrix while imposing STAT_NON_LINE (NEWTON = _F (REAC_ITER=0)).

2. In the case general, the tangent matrix is not-symmetrical. It is possible to do it thanksto the keyword SOLVEUR=_F (SYME = ‘YES’) of STAT_NON_LINE.

3. Concerning the nonlocal approach, the treatment of the boundary conditions is such asone could be brought, in the case of symmetrical structures, to treat the calculation ofthe whole of the structure and not of the “representative” part (cf [R5.04.02]).

4. The analytical expression of the tangent matrix is valid only for radial loadings (dr=d =0 ). In the other cases, the quadratic convergence of the method is not

guaranteed any more.

4.5 Stored internal variablesWe indicate in the table according to the internal variables stored in each point of Gauss for themodel of MAZARS :

Internal variable Physical direction

V1 D : variable of damage

V2 indicator of damage(0 so elastic, 1 if damaged i.e. as soon as D is not null any more)

V3 Tmax : temperature maximum reached at the point of gauss

V4 eq=〈 〉 :〈 〉 equivalent deformationTable 4.5-1: Stored internal variables.

5 Features and checking The law of behavior MAZARS, keyword BEHAVIOR of STAT_NON_LINE, associated materialMAZARS is usable in Code_Aster with various modelings:• classical version: 3D, D_PLAN, AXIS, C_PLAN (established analytical formulation, not to

use the method DEBORST)• not-local version: 3D_GRAD_EPSI , D_PLAN_GRAD_EPSI , C_PLAN_GRAD_EPSI , • coupled with the models of THHM (confer [R7.01.11]).

The law of MAZARS can be coupled with the model of creep BETON_UMLV_FP (confer[R7.01.06]) via the keyword KIT_DDI. This is true as well for the local version as thelocal one.

The law of behavior is checked by the following tests:



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COMP007b [V6.07.107]

Test of compression-dilation for study of the coupling thermics-cracking

HSNV129 [V7.22.129] Test of compression-dilation for study of the coupling thermics-crackingSSLA103 [V3.06.103] Calculation of the withdrawal of desiccation and the endogenouswithdrawal on a cylinderSSNP113 [V6.03.113] Rotation of the principal constraints (law of MAZARS)SSNP161 [V6.03.161] Biaxial tests of Kupfer SSNV157 [V6.04.157] Test of the method of delocalization per regularization of thedeformation on a variable bar of section in tractionSSNV169 [V6.04.169] Coupling creep-damageWTNV121 [V7.31.121] Damping of the concrete with a law of damage

Table 5-1 : existing CAS-tests

6 Bibliography[feeding-bottle1]

J. Mazars (1984). Application of the mechanics of the damage to the nonlinear behavior and therupture of the structural concrete, Doctorate of state of the University Paris VI.

[feeding-bottle2]

J. Lemaitre, J.L. Chaboche (1988), Mechanics of solid materials. ED. Dunod.

[feeding-bottle3]

J. Mazars, F. Hamon. With new strategy to formulate has 3D ramming model for concrete undermonotonic, cyclic and severe loadings. Engineering Structures, 2012, under review

[feeding-bottle4]

H. Askes (2000). Space Advanced discretization strategies for localised failure, mesh adaptivityand meshless methods, PhD thesis, Delft University of Technology, Faculty of Civil Engineeringand Geosciences.

[feeding-bottle5]

R.H.J. Peerlings, R. of Borst, W.A.M. Brekelmans, J.H.P of Vree, I. Spee (1996). Sumobservations one localization in nonroom and gradient ramming models, Eur. J. Mech. A/Solids,15 (6), 937-953.

[feeding-bottle6]

G. Pijaudier-Pooch, J. Mazars, J. Pulikowski (1991). Steel-concrete jump analysis with notroom continuous ramming, J. Structural Engrg ASCE, 117, 862-882.

[feeding-bottle7]

• C. Bellego, J.F. Dubé, G. Pijaudier-Pooch, B. Gerald (2003). Calibration of nonlocal rammingmodel from size effect tests, Eur. J. Mech. A/Solids, 22(1), 33-46.



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7 History of the versions of the documentVersion Aster

Author (S) orcontributor (S),organization

Description of the modifications

6.4 S.MICHEL-PONNELLE Initial text7.4 S.MICHEL-PONNELLE

9.4 S.MICHEL-PONNELLE,Marina BOTTONIAddition of the coupling UMLV-MAZARS + internal card11150 + 4th variable

11.0 Marina BOTTONI11.2 François HAMON Reformulation of the Mazars model


1 Introduction1.1 A law of behavior élasto-endommageable1.2 Limits of the local approach and methods of regularization1.3 Coupling with thermics1.4 Law of Mazars in the presence of a field of drying or hydration

2 Models of MAZARS2.1 Model of Origin of Mazars2.2 Revisited model of Mazars

3 Identification4 Digital resolution4.1 Evaluation of the internal variable Y4.2 Evaluation of the damage4.3 Calculation of the constraint4.4 Calculation of the tangent matrix4.5 Stored internal variables

5 Features and checking6 Bibliography7 History of the versions of the document

Documents

Model of damage of MAZARS - Code_Aster · 2019. 11. 23. · Code_Aster Version default Titre : Modèle d’endommagement de MAZARS Date : 25/09/2013 Page : 3/16 Responsable : HAMON